Question: Would you be able to explain this Qur’anic verse to me? 

Let no man guilty of adultery marry but a woman similarly guilty, or a polytheist; nor let any such a woman marry but a man similarly guilty or a polytheist; to the believers such a thing is forbidden. (24:3) 

    Does the above verse mean that a man or woman who has committed adultery in his or her life cannot marry at all? What happens to the single woman who is equally guilty? 
    Answer: Before I answer your question, let me point out that since Islam wants to build a society on the institution of family, it greatly protects this institution from everything that has the potential to disrupt it. Therefore, it prohibits fornication and adultery and regards them as punishable offences. It subscribes certain punishments for a man or a woman guilty of fornication or adultery. Part of this punishment is that such people cannot but marry either among themselves or among those who subscribe to idolatry. The wording of the verse clearly forbids marriage with chaste Muslims. 
    However, what needs to be clearly understood is that this punishment pertains only to fornicators and adulterers (both male and female) who have become liable to punishment once their crime has been proven by court evidence. This punishment is not to be given to people who have committed this crime but whose matter has not been bought before the court. As a matter of principle also, it needs to be appreciated that only the State is authorized to punish a person if it finds that he or she is guilty of a crime.
So if a married man who is proven guilty of adultery wants to marry again, he can then only marry an adulteress or a fornicator or a polytheist; similarly, the single lady who is proven to have committed adultery or fornication cannot marry a chaste believer. She can only marry an adulterer or a polytheist. 
    It also needs to be appreciated that the temperament of Islam is that in sins that do not relate to the violation of a party’s rights, the Almighty does not like that a criminal confess to his crime himself or that those who are aware of his crime report this matter to the authorities. If a person is of lewd and loose character, his misdoings may need to be reported, but if he has a morally sound reputation, Islam wants that even if he has faltered, his crime should be concealed and he should not be disgraced in the society. 
The Prophet (sws) is reported to have said: 

He among you who gets involved in such filth should hide behind the veil stretched out for him by Allah; but if he unfolds the veil, we shall implement the law of Allah upon him. (Mu’atta, Kitabu’l-Hudud)

    Similarly, he once told a person:

If you had hidden the crime of this [person], it would have been better for you. (Mu’atta, Kitabu’l-Hudud)